High School

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Example 4: Formulate the following system of linear equations.

At a college production of "Streetcar Named Desire," 400 tickets were sold. The ticket prices were [tex]\[tex]$8, \$[/tex]10[/tex], and [tex]\$12[/tex], and the total income from ticket sales was [tex]\$3700[/tex]. How many tickets of each type were sold if the combined number of [tex]\$8[/tex] and [tex]\$10[/tex] tickets sold was 7 times the number of [tex]\$12[/tex] tickets sold?

Multiple Choice:

a)
[tex]\[

\begin{array}{l}

x + y + z = 400 \\

8x + 10y + 12z = 3700 \\

7x + 7y = z

\end{array}

\][/tex]

b)
[tex]\[

\begin{array}{l}

x + y + z = 3700 \\

8x + 10y + 12z = 400 \\

x + y = 7z

\end{array}

\][/tex]

c)
[tex]\[

\begin{array}{l}

x + y + z = 400 \\

8x + 10y + 12z = 3700 \\

7x + y = z

\end{array}

\][/tex]

d)
[tex]\[

\begin{array}{l}

x + y + z = 400 \\

8x + 10y + 12z = 3700 \\

x + y = 7z

\end{array}

\][/tex]

Answer :

To formulate the system of linear equations based on the given conditions, let's consider each piece of information step-by-step.

### Given Information:
1. Total Number of Tickets Sold:
- There are three types of tickets: [tex]$8, $[/tex]10, and [tex]$12.
- The total number of tickets sold is 400.
- Let \( x \) represent the number of $[/tex]8 tickets.
- Let [tex]\( y \)[/tex] represent the number of [tex]$10 tickets.
- Let \( z \) represent the number of $[/tex]12 tickets.

So, we have the equation:
[tex]\[
x + y + z = 400
\][/tex]

2. Total Income from Ticket Sales:
- The number of [tex]$8 tickets multiplied by 8 gives the income from the $[/tex]8 tickets.
- The number of [tex]$10 tickets multiplied by 10 gives the income from the $[/tex]10 tickets.
- The number of [tex]$12 tickets multiplied by 12 gives the income from the $[/tex]12 tickets.
- The total income from all ticket sales is [tex]$3700.

So, we have the equation:
\[
8x + 10y + 12z = 3700
\]

3. Relationship Between the Number of Tickets Sold:
- The combined number of $[/tex]8 and [tex]$10 tickets sold (i.e., \( x + y \)) is 7 times the number of $[/tex]12 tickets sold (i.e., [tex]\( z \)[/tex]).

So, we have the equation:
[tex]\[
x + y = 7z
\][/tex]

### Formulating the System of Linear Equations:
Hence, the system of linear equations based on the given conditions is:

1. [tex]\( x + y + z = 400 \)[/tex]
2. [tex]\( 8x + 10y + 12z = 3700 \)[/tex]
3. [tex]\( x + y = 7z \)[/tex]

### Multiple Choice Identification:
Comparing the formulated equations with the provided choices, we find:

Option d:
[tex]\[
\begin{array}{l}
x + y + z = 400 \\
8x + 10y + 12z = 3700 \\
x + y = 7z
\end{array}
\][/tex]

This matches perfectly with our formulated system of equations.

Thus, the correct choice is:

d
[tex]\[
\begin{array}{l}
x + y + z = 400 \\
8x + 10y + 12z = 3700 \\
x + y = 7z
\end{array}
\][/tex]

Thanks for taking the time to read Example 4 Formulate the following system of linear equations At a college production of Streetcar Named Desire 400 tickets were sold The ticket prices were. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

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